Chap. 2. Polymers 2.1. Introduction - Polymers: synthetic materials <--> natural materials no gas phase, not simple liquid (much more viscous), not perfectly crystalline, etc
2.3. Polymer Chain Conformation 2.3.1. Isomerism - At a local scale, the polymer conformation depends on rotations about the bonds that make up the polymer backbone (for example C-C bonds). - trans and gauche + or gauche - conformations: some degree for oscillations under thermal fluctuations.
2.3.2. Molecular Size - Contour length: fully stretched (or extended), nl for a chain of n bonds of length l. - Dimensions of polymer coils: 1) rms end-to-end distance (average separation between chain ends): 2) radius of gyration (average distance of a chain segment from the center of mass of the coil):
- For random polymer (Gaussian) coils, and - For freely jointed chains, the valence angle model --> restriction on the bond angle -->
- For many polymer molecules with side groups that hinder rotation about, where = the steric factor *Stiffness of polymer chain: the characteristic ratio *For typical polymers, ranges between 1.5 and 2.5 and lies bet 5 and 10. - Excluded volume effect: exclusion by occupied segments -> self-avoiding walks where = the expansion factor. 2.4. Characterization - Polymerization process: the probability of attachment of a monomers (NOT monodisperse, distribution of mass) the number and weight average molar mass: and --> the polydispersity index
[Reading Assignment] 2.4.2, 2.4.3, 2.4.4 2.5. Polymer Solutions 2.5.1. Solvent Quality: Good, Bad, and Theta - In dilute solution, the conformation of a polymer chain depends on the interactions bet chain segments and solvent molecules. *chain segment-solvent and segment-segment interactions 1) Good solvent: a chain expands from its unperturbed dimensions to maximize the number of segment-solvent contacts -> the coil is swollen. 2) Poor solvent: the chain contacts to minimize interactions with solvent. *Excluded volume effect: chains expands to reduce unfavorable segment-segment interactions 3) Theta solvent: two effects are perfectly balanced -> polymer adopts unperturbed dimensions. the mean square end-to-end distance:
- In a good solvent, with for a Gaussian chain (expanded coil) and in a theta solvent, for an unperturbed chain *Exact value of the exponent (the Flory exponent). - The ratio bet perturbed and unperturbed dimensions: the expansion factor good solvent ( ), theta solvent ( ), poor solvent ( ) *The solvent molecules will change the excluded volume for a polymer coil; how much volume it takes up and prevents neighboring chains from occupying. *In a theta solvent, the excluded volume is zero -> the solvent is ideal! *The second virial coefficient depends on interactions between pairs of molecules, proportional to the excluded volume. *Theta conditions are attained at the theta temperature depending on the solvent.
2.5.2. Concentration Regimes - Interactions bet polymer molecules in solution depend strongly on concentration. In a dilute solution, the molecules are well separated on average -> no interactions. At the coil overlap concentration ( ), the coils are just in contact. => the concentration of polymers in solution is equal to the average concentration of segments in an individual coil; with V = vol of a coil
- Using the scaling relation,, the concentration In a good solvent, ; in a theta solvent, - In terms of the polymer volume fraction, for in the absence of the solvent. ( = the overlap volume fraction) 2.5.4. Coil-Globule Transition - Contraction of the coil: segment-segment attractions, not repulsions. Attractions bet monomers leads to a collapse of the coil into a compact globule conformation (analogous to the condensation of a liquid from a gas). In the swollen coil, the segments are, on average, well separated but in the globule, the density of segments is high. - For a globule, dimensions scale with (for a Gaussian coil, )
2.5.6. Flory-Huggins Theory - Mixing of polymer with solvent <--> mixing of two liquids of small molecules - Thermodynamic requirement for miscibility in terms of the molar Gibbs free energy: with = the molar enthalpy (heat) of mixing = the molar entropy of mixing For an ideal solution, the enthalpy change on mixing. - For a mixture of system 1 and system 2,. - For most polymers, 1) finite heat of mixing when dissolved in a low molecular weight solvent (non-idela). 2) a large change in entropy on mixing that results from volume changes: The segment of the polymer molecule are constrained to be connected into a chain. ==> and
- Lattice model for calculation of the configurational entropy: 1) Entropy change on mixing polymer with solvent: solvent molecule 1, polymer molecule 2 with r segments (connected)
- Calculation of configurational entropy: Under the assumption that solvent molecules and polymer chains lie in on the lattices (allowing for the connectivity of the chain), it is needed to calculate the number of possible arrangements - The molar entropy change on mixing is given by (see, Cowie, 1991 for details). with = the fraction of sites occupied. 2) The change in enthalpy arising from the formation of solvent-polymer contacts (1-2) at the expense of solvent-solvent (1-1) and polymer-polymer contacts (2-2): In the pseudo-chemical reaction, The associated energy change for breaking (1-1) and (2-2) contacts and forming (1-2) contacts is with = the contact energy.
- Under no volume change on mixing, the internal energy change is equivalent to the enthalpy change: For the formation of new contacts, Assuming random mixing and denoting the coordination number of the lattice by z, each polymer chain is surrounded by solvent molecules: then, for polymer molecules, Using the relation, - In terms of a dimensionless parameter (a measure of the interaction enthalpy per solvent molecule), with = the (Flory-Huggins) interaction parameter. - The second virial coefficient with = the partial specific volume of the polymer and = the molar volume of the solvent in solution. ; theta condition (, ideal solution) (poor solvent) and (good solvent)
- The Flory-Huggins Equation for the Gibbs free energy <Assumptions>: often NOT valid 1) No volume change upon mixing polymer and solvent 2) No contribution of the chain flexibility to the entropy (lattice model) 3) Neglect of specific solvent-polymer interactions (for example, polar interactions and hydrogen bonding) 4) Internal inconsistency: for randomized solvent-segment contacts 5) Simplification of the interaction parameter : need of both entropic and enthalpic contribution, concentration dependence ==> a reasonable first approximation to the thermodynamic of many polymer mixtures!
2.5.7. Critical Solution Temperatures - For the phase separation in polymer-solvent or polymer-polymer mixtures, with B whose sign determines whether the mixing is favored enthalpically at high temperature or low temperature. - A blend is completely miscible if for a blend is less than that of the components A composition for which is greater than that of two coexisting phases, the phase separation into two phases will occur. - Phase behavior of a binary blend of polymer-solvent or polymer-polymer: At hight temperature, for a mixture is always less than that for the pure components, and then a homogeneous (mixed) phase is stable. As temperature is reduced, the homogeneous mixture becomes unstable, and a phase separation occurs.
- The compositions of two phases and : NOT corresponding to the minima in but defined by the points of contact of the double tangent line CC'
- Binodal curve: the locus of the compositions at the tangent points as a function of temperature -> a critical solution temperature. For a polymer-solvent mixture, the binodal is called as a cloud point curve. (two coexisting phases with droplets of one phase dispersed in the other) - The limit of the thermodynamic stability: ; inflection points - Spinodal curve: the locus of the inflection points (for the thermodynamic stability) *Phase separation occurs spontaneously for all compositions between those defining the spinodal curve.
*In the region bet the spinodal and the binodal curve, the homogeneous phase is metastable. - Oswald ripening: phase separation leads to a bicontinuous structure, the system may coarsen, and eventually the bicontinuous structure can evolve into a droplet structure as the system seeks to minimize the free energy associated with creation of an interface between phase separated domains.
- Upper or lower critical solution temperature (UCST or LCST): The two sides of the spinodal and binodal curves meet at a common maximum or minimum.
2.6. Amorphous Polymers 2.6.3. Dynamics - The molar mass dependence of the viscosity of a polymer melt *Increase of entanglements
- The Rouse model: the polymer chain is supposed to comprise a series of sequences, each of which is sufficiently long to have a Gaussian conformation. *Each sequence consists of a bead and a spring: the frictional force due to the viscous medium acts on the beads while the spring is used to model the elastic behavior of the polymer chain. * Valid for describing the dynamics of polymer chains at intermediate time-scales the scaling of relaxation time, diffusion coefficient, viscosity - The reptation theory: diffusion of a polymer chain through a tube defined by surrounding entangled chains.
2.7. Crystalline Polymers 2.7.2. Hierarchical Structure: Molecular level (extended or folded) -> arrangement in layers -> superstructures
[Reading Assignment] 2.8, 2.9, 2.10 2.11. Polymer Blends and Block Copolymers - Different functionalities: one component may be glassy or crystalline, giving strength, while another may be rubbery, providing flexibility and processibility. - Different structures are stable for block coploymers with different compositions, specified as the volume fraction of one block, f. - Microphase separation, order-disorder transition
- Phase diagram parameterized by f and with = the Flory-Huggins parameter and = the degree of polymerization for symmetric diblock coploymers
2.14. Electronic and Opto-electronic Polymers - Conjugated polymers: conducting, semiconducting, light-emitting